Subgroups — C₁₀₉×#3511 #28360765

Order 13952.

1104 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#555×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#554×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#553×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	#548×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	#547×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#542×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	#537×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#533×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#532×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#520×C₁₀₉	C₂	1	2	yes	maximal, complemented
14	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
17	C₂×#41×C₁₀₉	C₂²	1	4	yes	—
18	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
22	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#127×C₁₀₉	C₂²	1	1	yes	characteristic
25	#126×C₁₀₉	C₂²	1	8	yes	—
26	#125×C₁₀₉	C₂²	1	4	yes	—
27	#124×C₁₀₉	C₂²	1	2	yes	—
28	#124×C₁₀₉	C₂²	1	4	yes	—
29	#124×C₁₀₉	C₂²	1	8	yes	—
30	#123×C₁₀₉	C₂²	1	8	yes	—
31	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
32	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
33	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
34	C₂²⋊Q₈×C₁₀₉	C₂²	1	8	yes	—
35	#121×C₁₀₉	C₂²	1	4	yes	—
36	#121×C₁₀₉	C₂²	1	2	yes	—
37	#121×C₁₀₉	C₂²	1	2	yes	—
38	#121×C₁₀₉	C₂²	1	4	yes	—
39	#121×C₁₀₉	C₂²	1	4	yes	—
40	#121×C₁₀₉	C₂²	1	8	yes	—
41	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
44	C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
45	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
46	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
47	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
48	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
49	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
50	C₄×D₄×C₁₀₉	C₂²	1	8	yes	—
51	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
52	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
53	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
54	#117×C₁₀₉	C₂²	1	1	yes	characteristic
55	#117×C₁₀₉	C₂²	1	2	yes	—
56	#117×C₁₀₉	C₂²	1	4	yes	—
57	#117×C₁₀₉	C₂²	1	4	yes	—
58	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
59	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
60	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
61	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
62	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
63	#41×C₁₀₉	—	2	4	no	—
64	#41×C₁₀₉	—	2	8	no	—
65	#41×C₁₀₉	—	4	8	no	—
66	C₂×Q₈×C₁₀₉	—	4	4	no	—
67	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
69	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
70	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
71	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
72	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
73	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
74	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
76	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
77	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
78	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
83	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
84	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
85	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
86	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
89	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
90	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
91	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
92	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
93	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
95	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
96	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
97	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
98	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
99	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
100	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
101	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
102	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
103	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
104	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
105	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
106	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₄²×C₁₀₉	C₂³	1	2	yes	—
109	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₄²×C₁₀₉	C₂³	1	2	yes	—
111	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
112	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
113	C₄²×C₁₀₉	C₂³	1	2	yes	—
114	C₄²×C₁₀₉	C₂³	1	8	yes	—
115	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
116	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
117	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	Q₈×C₁₀₉	—	2	2	no	—
119	Q₈×C₁₀₉	—	2	4	no	—
120	Q₈×C₁₀₉	—	2	8	no	—
121	Q₈×C₁₀₉	—	4	4	no	—
122	D₄×C₁₀₉	—	2	4	no	—
123	D₄×C₁₀₉	—	2	8	no	—
124	D₄×C₁₀₉	—	2	2	no	—
125	D₄×C₁₀₉	—	2	4	no	—
126	D₄×C₁₀₉	—	2	4	no	—
127	D₄×C₁₀₉	—	2	4	no	—
128	D₄×C₁₀₉	—	4	8	no	—
129	C₂×C₄×C₁₀₉	—	2	8	no	—
130	C₂×C₄×C₁₀₉	—	2	4	no	—
131	C₂×C₄×C₁₀₉	—	2	2	no	—
132	C₂×C₄×C₁₀₉	—	2	4	no	—
133	C₂×C₄×C₁₀₉	—	2	2	no	—
134	C₂×C₄×C₁₀₉	—	2	4	no	—
135	C₂×C₄×C₁₀₉	—	4	4	no	—
136	C₂×C₄×C₁₀₉	—	4	8	no	—
137	C₂×C₄×C₁₀₉	#41	1	4	yes	complemented
138	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
139	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
140	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
141	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
142	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
143	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
144	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
145	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
146	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
147	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
148	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
149	C₂²×C₁₀₉	—	2	1	no	—
150	C₂²×C₁₀₉	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	2	no	—
152	C₂²×C₁₀₉	—	2	4	no	—
153	C₂²×C₁₀₉	—	2	2	no	—
154	C₂²×C₁₀₉	—	2	4	no	—
155	C₂²×C₁₀₉	—	2	2	no	—
156	C₂²×C₁₀₉	—	2	4	no	—
157	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
158	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
159	C₄×C₁₀₉	—	2	4	no	—
160	C₄×C₁₀₉	—	2	2	no	—
161	C₄×C₁₀₉	—	2	2	no	—
162	C₄×C₁₀₉	—	2	1	no	—
163	C₄×C₁₀₉	—	2	8	no	—
164	C₄×C₁₀₉	—	2	2	no	—
165	C₄×C₁₀₉	—	2	1	no	—
166	C₄×C₁₀₉	C₂×#41	1	2	yes	—
167	C₄×C₁₀₉	C₂×#41	1	2	yes	—
168	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
169	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
170	C₂×C₁₀₉	—	2	2	no	—
171	C₂×C₁₀₉	—	4	2	no	—
172	C₂×C₁₀₉	—	4	4	no	—
173	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
174	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
176	#3511	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₁₀₉	#3511	1	1	yes	characteristic, direct product factor, Sylow
178	#556	C₂×C₁₀₉	1	2	yes	complemented
179	#555	C₂×C₁₀₉	1	2	yes	complemented
180	#554	C₂×C₁₀₉	1	2	yes	complemented
181	#553	C₂×C₁₀₉	1	4	yes	complemented
182	#548	C₂×C₁₀₉	1	4	yes	complemented
183	#547	C₂×C₁₀₉	1	2	yes	complemented
184	#542	C₂×C₁₀₉	1	4	yes	complemented
185	#537	C₂×C₁₀₉	1	2	yes	complemented
186	#533	C₂×C₁₀₉	1	2	yes	complemented
187	#532	C₂×C₁₀₉	1	1	yes	characteristic, complemented
188	#532	C₂×C₁₀₉	1	2	yes	complemented
189	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
190	#520	C₂×C₁₀₉	1	2	yes	complemented
191	C₄×#41	C₂×C₁₀₉	1	1	yes	characteristic, complemented
192	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
193	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
194	C₂×#41	C₂²×C₁₀₉	1	4	yes	—
195	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
196	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
197	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
198	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
199	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
200	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
201	#127	C₂²×C₁₀₉	1	1	yes	characteristic
202	#126	C₂²×C₁₀₉	1	8	yes	—
203	#125	C₂²×C₁₀₉	1	4	yes	—
204	#124	C₂²×C₁₀₉	1	2	yes	—
205	#124	C₂²×C₁₀₉	1	4	yes	—
206	#124	C₂²×C₁₀₉	1	8	yes	—
207	#123	C₂²×C₁₀₉	1	8	yes	—
208	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
209	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
210	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
211	C₂²⋊Q₈	C₂²×C₁₀₉	1	8	yes	—
212	#121	C₂²×C₁₀₉	1	4	yes	—
213	#121	C₂²×C₁₀₉	1	2	yes	—
214	#121	C₂²×C₁₀₉	1	2	yes	—
215	#121	C₂²×C₁₀₉	1	4	yes	—
216	#121	C₂²×C₁₀₉	1	4	yes	—
217	#121	C₂²×C₁₀₉	1	8	yes	—
218	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
219	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
220	C₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
221	C₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
222	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
223	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
224	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
225	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
226	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
227	C₄×D₄	C₂²×C₁₀₉	1	8	yes	—
228	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
229	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
230	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
231	#117	C₂²×C₁₀₉	1	1	yes	characteristic
232	#117	C₂²×C₁₀₉	1	2	yes	—
233	#117	C₂²×C₁₀₉	1	4	yes	—
234	#117	C₂²×C₁₀₉	1	4	yes	—
235	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
236	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
237	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
238	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
239	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
240	#41	—	2	4	no	—
241	#41	—	2	8	no	—
242	#41	—	4	8	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
245	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
246	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
247	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
248	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
249	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
250	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
251	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
252	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
253	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
254	C₂×D₄	C₂³×C₁₀₉	1	8	yes	—
255	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
256	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
257	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
258	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
259	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
260	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
261	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
262	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
263	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
266	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
267	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
268	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
269	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
270	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
272	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
273	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
274	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
275	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
276	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
277	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
278	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
279	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
280	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
281	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
282	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
283	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
284	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
285	C₄²	C₂³×C₁₀₉	1	2	yes	—
286	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
287	C₄²	C₂³×C₁₀₉	1	2	yes	—
288	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
289	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
290	C₄²	C₂³×C₁₀₉	1	2	yes	—
291	C₄²	C₂³×C₁₀₉	1	8	yes	—
292	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
293	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
294	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
295	Q₈	—	2	2	no	—
296	Q₈	—	2	4	no	—
297	Q₈	—	2	8	no	—
298	Q₈	—	4	4	no	—
299	D₄	—	2	4	no	—
300	D₄	—	2	8	no	—
301	D₄	—	2	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	2	4	no	—
304	D₄	—	2	4	no	—
305	D₄	—	4	8	no	—
306	C₂×C₄	—	2	8	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	8	no	—
314	C₂×C₄	#41×C₁₀₉	1	4	yes	complemented
315	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
316	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
317	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
318	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
319	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
320	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
321	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
322	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
323	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
324	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
325	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	4	no	—
334	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
335	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
336	C₄	—	2	4	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	8	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	C₂×#41×C₁₀₉	1	2	yes	—
344	C₄	C₂×#41×C₁₀₉	1	2	yes	—
345	C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
346	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
347	C₂	—	2	2	no	—
348	C₂	—	4	2	no	—
349	C₂	—	4	4	no	—
350	C₂	C₂²×#41×C₁₀₉	1	1	yes	characteristic
351	C₂	C₂×H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
352	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #556×C₁₀₉	C₂	semidirect
2. #555×C₁₀₉	C₂	semidirect
3. #554×C₁₀₉	C₂	semidirect
4. #553×C₁₀₉	C₂	semidirect
5. #548×C₁₀₉	C₂	semidirect
6. #547×C₁₀₉	C₂	semidirect
7. #542×C₁₀₉	C₂	semidirect
8. #537×C₁₀₉	C₂	semidirect
9. #533×C₁₀₉	C₂	semidirect
10. #532×C₁₀₉	C₂	semidirect
11. #532×C₁₀₉	C₂	semidirect
12. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
13. #520×C₁₀₉	C₂	semidirect
14. C₄×#41×C₁₀₉	C₂	semidirect
137. C₂×C₄×C₁₀₉	#41	semidirect
176. #3511	C₁₀₉	direct (N × H)
177. C₁₀₉	#3511	direct (N × H)
178. #556	C₂×C₁₀₉	semidirect
179. #555	C₂×C₁₀₉	semidirect
180. #554	C₂×C₁₀₉	semidirect
181. #553	C₂×C₁₀₉	semidirect
182. #548	C₂×C₁₀₉	semidirect
183. #547	C₂×C₁₀₉	semidirect
184. #542	C₂×C₁₀₉	semidirect
185. #537	C₂×C₁₀₉	semidirect
186. #533	C₂×C₁₀₉	semidirect
187. #532	C₂×C₁₀₉	semidirect
188. #532	C₂×C₁₀₉	semidirect
189. C₂×C₄⋊Q₈	C₂×C₁₀₉	semidirect
190. #520	C₂×C₁₀₉	semidirect
191. C₄×#41	C₂×C₁₀₉	semidirect
314. C₂×C₄	#41×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#556×C₁₀₉	2
#555×C₁₀₉	2
#554×C₁₀₉	2
#553×C₁₀₉	2
#548×C₁₀₉	2
#547×C₁₀₉	2
#542×C₁₀₉	2
#537×C₁₀₉	2
#533×C₁₀₉	2
#532×C₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
#520×C₁₀₉	2
C₄×#41×C₁₀₉	2
#3511	109

Subgroups — C109×#3511 #28360765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3511 #28360765